Torque free motion

The simulation displays (in the space frame on the left and in the body frame on the right) the evolution of a torque free ellipsoid around its center of mass. The Lagrangian is as follows:
L = ½ I_x ω_x² +½ I_y ω_y² + ½ I_z ω_z² ,
where the components of the angular velocity in the body frame are
ω_x = φ' sin θ sin ψ + θ' cos ψ
ω_y = φ' sin θ cos ψ - θ' sin ψ
ω_z = φ' cos θ + ψ'
(we use ' for the time derivative).

To see the definition of the Euler angles (θ,φ,ψ), click on Variables in the simulation below.
There you can also select the initial values, as well as the principal moments of inertia I_x, I_y and I_z.
Units are arbitrary.

Check Velocities to see the evolution of the angular velocity components in the body frame (ω_x,ω_y,ω_z) and in the space frame (ω'_x,ω'_y,ω'_z), where.
ω'_x = θ' cos φ + ψ' sin φ sin θ
ω'_y = θ' sin φ - ψ' cos φ sin θ
ω'_z = φ' + ψ' cos θ
The point of view of the three-dimensional projections can be changed with the cursors or the mouse.
Each whole image can be moved with the mouse while pressing Ctrl.
To change the zoom in the projections, press Shift when moving up or down the mouse pointer.
Put the mouse pointer over an element to get information about it.

Reference: H. Goldstein, Ch. Poole and J. Safko, Classical Mechanics, 3^rd edition, Addison-Wesley, San Francisco

Activities

Press Start and get familiarized with the display options below the images.
Let us check the Tennis Racket Theorem:
- Make sure the body initially spins nearly around its Z axis, i.e., select a fairly higher value for ψ' than for θ' and φ'.
- Check what happens in the three cases in which
  - I_z < I_x, I_y
  - I_z > I_x, I_y
  - I_x < I_z < I_y or I_x > I_z > I_y
- Discuss the results.
- When the initial motion is unstable, is it periodic the evolution in the space/body frame? (It may help enabling Trajectories and/or Velocities.) Comment the result of your numerical experience.

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.